Problem 22 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(x y^{3}\right) $$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \( \log_b(xy^3) \) is \( \log_b(x) + 3\log_b(y) \).

Step 1: Recall the Product Rule of Logarithms

Remember the product rule of logarithms which states that \( \log_b(xy) = \log_b(x) + \log_b(y) \). This rule allows us to separate a single logarithm of a product into a sum of multiple logarithms.

Step 2: Apply the Product Rule

Applying this rule, we can rewrite the given logarithm \(\log_b(xy^3)\) as \(\log_b(x) + \log_b(y^3)\).

Step 3: Recall the Power Rule of Logarithms

Now remember the power rule of logarithms that states \( \log_b (a^n) = n\log_b(a) \). This rule allows us to separate the exponent from the argument of the logarithm and bring it out front as a coefficient.

Step 4: Apply the Power Rule

Applying this rule to our expression, we can rewrite the second term \( \log_b(y^3) \) as \( 3\log_b(y) \). So, the expanded form is \( \log_b(x) + 3\log_b(y) \).